The physical nature of spacetime

[jimbobjames]

I asked a similar question previously, but my enthusiasm got the better of me and I may have smothered it with too many tangential questions. So I will try again.

The statements that "spacetime is curved by matter" and that "curved spacetime tells matter how to move" summarize GR succinctly. But what exactly is this entity which we label "spacetime" and which gets curved by matter and energy?

Think of a test particle in a vacuum - what is the nature of the physical entity which is in direct contact with that particle, the curvature of which is caused by matter and energy?

Is it:

a) A fluid whose physics we have not yet understood?
b) A sea of gravitons (which we hope to observe at the LHC very soon)?
c) An entity which lives in dimensions we cannot (yet) observe?
d) We don't know - the best description we have is the mathematical model for it - a "Pseudo-Riemannian 4-D manifold with a metric"
e) Something else?

Thanks for your help with this.

 

[pervect]

As far as GR is concerned, the best answer is d), a 4-d semi-Riemanian manifold with a Lorentzian metric.

Some forms of string theory might take viewpoint 'c'

Some forms of quantum theory might take viewpoint 'b'. (I'm not sure if there is anything of this form that works well, though).

But GR takes view 'd'.

You might want to think about "how could we know"? Science tells us that we look for agreement with experiment. So GR takes viewpoint d, and a few other assumptions, and tries to make predictions from it. It works pretty well, but we have reason to believe it's not the complete answer.

Other viewpoints are certainly possible, and if they yield self-consistent answers that are compatible with experiment, they can be studied scientifically.

 

[jimbobjames]

Thanks a lot. That is very helpful.

I suppose the issue I would have with answer d) would be that it directly equates a mathematical entity (a manifold) with something physical (spacetime). Physics = Maths. But surely maths provides the tools to help us model and better understand the physics, rather than actually being the physics?

Unless of course we are saying that spacetime is NOT itself a physical entity but rather a mathematical tool - ie its exists only insofar as it describes separations (intervals) between real physical 4-D entities (events).

A related but more specific question might be: What is it that directly interacts with a clock which is close to the Earth's surface (ie low in a weak field where only the time curvature is significant) and causes it to tick more slowly than a clock which is higher up?

About your question: "How could we know"? I suppose only by trying to improve on the existing theories and coming up with clever experiments to check the predictions. Perhaps even logically consistent thought experiments will do in the first instance.

Maybe I just need to understand that matter (in fact everything) is really made up of a field - what we perceive to be solid is really an illusion created by our minds interacting with the field. It will then (hopefully someday) be somewhat easier for me to imagine a gravitational field interacting with this matter field and vice versa in such a way as to produce the effects we call gravity. In other words, instead of me trying to promote the gravitational field to something I can perceive (like matter I am used to dealing with), I need to understand that matter is not really what it looks like but rather that it is actually made up of fields itself.

Thanks again.

 

[pmb_phy]

I asked a similar question previously, but my enthusiasm got the better of me and I may have smothered it with too many tangential questions. So I will try again.

The statements that "spacetime is curved by matter" and that "curved spacetime tells matter how to move" summarize GR succinctly. But what exactly is this entity which we label "spacetime" and which gets curved by matter and energy?

Think of a test particle in a vacuum - what is the nature of the physical entity which is in direct contact with that particle, the curvature of which is caused by matter and energy?

Is it:

a) A fluid whose physics we have not yet understood?
b) A sea of gravitons (which we hope to observe at the LHC very soon)?
c) An entity which lives in dimensions we cannot (yet) observe?
d) We don't know - the best description we have is the mathematical model for it - a "Pseudo-Riemannian 4-D manifold with a metric"
e) Something else?

Thanks for your help with this.

I assume that you understand, or at least have an intuitive idea of, what space and time are seperately. Each corresponds to something in the physical. These are things concieved by humans to describe what they observer in nature. Now consider a particular place at a particular time. E.g. Times Square, NY at 11:59pm December 31, 2007. This is what we call an event. In this case the event is one minute before the date turns to Jan 1, 2008. If we had a spacetime diagram, in which we plot time vs space then the (ct,x) plane would consist of all events which happens on the line which consists of y= z = 0. The events (ct,x) lie in a plane and that place is called the subspace of spacetime (ct,x). Now suppose we consider all of space and all of time. Then what we're looking at looks locally like R⁴. If a region of spacetime looks like this then that region is (loosley) called a spacetime manifold. Now keep in mind that this is a mathematical entity which corresponds/explain/describes nature. Such a manifold (the mathematical thingy) can be curved like the surface of a baseball is curved (i.e. curvature refers to intrinsic curvature). But to define curvature one must either define an affine connection on the manifold or a metric field.

Does that help?

Best wishes

Pete

 

[cesiumfrog]

Maybe I just need to understand that matter (in fact everything) is really made up of a field - what we perceive to be solid is really an illusion created by our minds interacting with the field. It will then (hopefully someday) be somewhat easier for me to imagine a gravitational field interacting with this matter field and vice versa in such a way as to produce the effects we call gravity. In other words, instead of me trying to promote the gravitational field to something I can perceive (like matter I am used to dealing with), I need to understand that matter is not really what it looks like but rather that it is actually made up of fields itself.

If you forget gravity and go right back to what you're most comfortable with: solid tangible particles that interact in a completely traditional manner (with an absolute/non-relativistic flow of time, and three Cartesian spatial dimensions).. do you still feel compelled to say that the space itself (in Newtonian mechanics) is a separate tangible physical entity? To me that would seem silly, and so you should appreciate the silliness even if the geometry rules are not necessarilly trivial.

And sure, approaching it from the other end, modern physics constantly teaches us that the traditional intuition (of expecting things to be describable in terms of tangible particles) is fundamentally wrong.

 

[pervect]

Thanks a lot. That is very helpful.

I suppose the issue I would have with answer d) would be that it directly equates a mathematical entity (a manifold) with something physical (spacetime). Physics = Maths. But surely maths provides the tools to help us model and better understand the physics, rather than actually being the physics?

Unless of course we are saying that spacetime is NOT itself a physical entity but rather a mathematical tool - ie its exists only insofar as it describes separations (intervals) between real physical 4-D entities (events).

The way I see it, your original question is mainly philosophical. Philosophical questions generally don't have any definite answers, at least not that can be found by experiments. So, the way I see it, you shouldn't really expect to get a definite answer to your questions.

Science can be defined by the fact that it focuses mainly on experimental results.

While this is somewhat limiting, by narrowing the field to things we can actually measure, science gains more from the narrow focus than it loses, at least in terms of useful discoveries.

As I tried to point out, the two disciplines aren't totally disconnected, it's a matter of emphasis. Philosophy can be important in the structures needed to even formulate new theories. Interestingly enough, though, it seems to me that mathematics is the source of more inspiration than philosophy is. To paraphrase a half-remembered remark, if a million philosophers sat down at a million typewriters and typed for a thousand years, they still wouldn't come up with anything as bizarre as quantum mechanics. (Which, in spite of its bizarreness, works very very well.)

I wouldn't underestimate the importance of finding out some of the mathematical properties that space-time has, either. While it may not be a complete set of all the properties, knowing some of the properties of this entity is very helpful.

Of course, it may be useful to point out that while at the human scale in we live in we have clocks and rulers and the geometrical structure that goes with them (i.e. the manifold structure we talked about earlier), there is some reason to suspect that space-time does not work in this manner at very small scales. (See for instance Wheeler's "quantum foam".) Unfortunately for us, some of the interesting scales are experimentally inaccessible, which limits what we can really discover about how they work. This is one of the things that makes quantum gravity very difficult.

A related but more specific question might be: What is it that directly interacts with a clock which is close to the Earth's surface (ie low in a weak field where only the time curvature is significant) and causes it to tick more slowly than a clock which is higher up?

Can you think of an experiment that would help answer this question? If you can, it's a scientific question. If you can't, it's a philosophical question - or at least, it can be presumed to be philospohical, until such a time as you come up with a way to resolve the question by some sort of experiment, at which point it again belongs to the realm of science.

I find that it's very helpful to be able to come up with such thought experiments, even if the experiments are not currently practical to carry out.

 

[jimbobjames]

@Pete: Thanks. I didnt mention this, but I have studied GR in my spare time and would say I have a fairly good grasp of the mathematics. Perhaps my favorite introduction to GR is that by Dirac who gets straight to the point - I think I was pleasantly surprised that I could follow his derivations.

@cesiumfrog: "do you still feel compelled to say that the space itself (in Newtonian mechanics) is a separate tangible physical entity? "

It is GR, not me, that is telling us that spacetime is curved. And by saying that, is it not implicitly saying that spacetime is a thing which can be curved?

@Pervect: "The way I see it, your original question is mainly philosophical."

GR is saying that spacetime is curved. The question is - what is meant by spacetime in that context? I think that is a perfectly valid question, not at all unscientific.

"I wouldn't underestimate the importance of finding out some of the mathematical properties that space-time has, either."

I totally agree. Through mathematics, Dirac's equation predicted the positron, and Einstein's field equation predicted an expanding universe, even if he refused to believe it himself. And although we have trouble understanding (or "picturing") wave-particle duality, mathematics provides objects which can simultaneously embody both wave and particle features. It is as if Nature was speaking to us through mathematics.

"Can you think of an experiment that would help answer this question? "

Here is a simple thought experiment. Let go of a test body (say an apple) over the surface of a planet - assume there is no athmosphere (vacuum). Observe what happens.
Result: It accelerates relative to the ground and you can measure the acceleration.

If we postulate that there is nothing in direct contact with the apple (ie no physical entity called spacetime) then we are forced to believe that there is some kind of an action at a distance which violates special relativity.

So we conclude from this simple thought experiment that there is in fact something in direct contact with the apple. I'd like to understand that something a bit better.

And thanks pervect! You already did answer this partially when you wrote that b) and c) above are related to some string and quantum theories and I may find more answers there. It sounds like GR itself has little to say about the question I am asking and I need to study quantum field theories (!).

 

[masudr]

The question is - what is meant by spacetime in that context? I think that is a perfectly valid question, not at all unscientific.

Unscientific here (as used by Pervect) has a very specific meaning. i.e. in the Popperian sense: if a premise cannot have testable predictions, then it is an unscientific premise. So if you can think up perhaps a prediction that would distinguish spacetime from "being something tangible" and the converse, then yes, such a premise would be scientific. Otherwise, it is unscientific.

I may be mistaken, but it appears your usage of the term unscientific is more about "not regarding science," and as you say, if we use this definition the basis of your discussion is scientific.

The whole point of the Popperian distinction is to try and work out which hypotheses we can determine the truth of by experiment (and hence understand a truth of the universe) i.e. scientific, and those that we can't.

 

[yogi]

"A related but more specific question might be: What is it that directly interacts with a clock which is close to the Earth's surface (ie low in a weak field where only the time curvature is significant) and causes it to tick more slowly than a clock which is higher up?"

Even more puzzling is why two clocks in a uniform acceleration field run at different rates. Where there is a difference in the gravitational force (as is the case near the earth) it would seem there is some physical bases for the phenomena. But where the front clock in a rocket and the rear clock are subjected to identical accelerations, how does the lower clock know it is the lower clock, and vice versa. Seems these are problems that cannot be attacked by reductionism - perhaps a more holistic approach is called for.

pervect: "Can you think of an experiment that would help answer this question? If you can, it's a scientific question. If you can't, it's a philosophical question - or at least, it can be presumed to be philospohical, until such a time as you come up with a way to resolve the question by some sort of experiment, at which point it again belongs to the realm of science.

I find that it's very helpful to be able to come up with such thought experiments, even if the experiments are not currently practical to carry out."

This hints of something you might have in mind - care to share

 

[cesiumfrog]

It is GR, not me, that is telling us that spacetime is curved. And by saying that, is it not implicitly saying that spacetime is a thing which can be curved?

My point is that pre-GR theories told us space was geometrically flat everywhere (flatness being just one of among all the mathematically conceivable geometries), and yet we didn't get all excited then about space itself being a "thing".

Here is a simple thought experiment. [..] If we postulate that there is nothing in direct contact with the apple (ie no physical entity called spacetime) then we are forced to believe [since the apple accelarates toward the ground] that there is some kind of an action at a distance which violates special relativity.

You've misunderstood both GR and SR; Not only is there no "action" on the apple, but it wouldn't make a difference if there was:

- SR prohibits only a specific type of "action at a distance", so even if GR worked as you describe (ie., just forget the equivalence principle, let gravity "act" almost like Newtonian gravity except impose that mass-rearrangements won't instantly change the force on distant bodies) it still wouldn't violate SR. (The problem, which specifically is with causality, would exist only if gravitational waves were to go faster than the speed of light.)

- According to GR, the apple does not accelerate towards the ground. It is the ground accelerating toward the apple (and the ground is being accelerated by the Earth beneath and in direct contact with it). [In an inconsistent kind of way, you can investigate this by putting an accelerometer on the earth, and another on the apple..]

 

[pervect]

@Pervect: "The way I see it, your original question is mainly philosophical."

GR is saying that spacetime is curved. The question is - what is meant by spacetime in that context? I think that is a perfectly valid question, not at all unscientific.

In that context, you can consider space-time to be a 4-d manifold. (More precisely a semi-Riemanain manifold with a Lorentz metric as was mentioned earlier).

Space-time diagrams in special relativity are drawn on a flat sheet of paper. Hopefully it is obvious that in this paper depiction, the time dimension iis usually represented as a spatial dimension, and that a couple of spatial dimensions have been supressed.

By switching to a manifold structure, GR is basically suggesting that we draw our space-time diagrams on curved surfaces rather than on flat sheets of paper.

Here is a simple thought experiment. Let go of a test body (say an apple) over the surface of a planet - assume there is no athmosphere (vacuum). Observe what happens.
Result: It accelerates relative to the ground and you can measure the acceleration.

If we postulate that there is nothing in direct contact with the apple (ie no physical entity called spacetime) then we are forced to believe that there is some kind of an action at a distance which violates special relativity.

So we conclude from this simple thought experiment that there is in fact something in direct contact with the apple. I'd like to understand that something a bit better.

And thanks pervect! You already did answer this partially when you wrote that b) and c) above are related to some string and quantum theories and I may find more answers there. It sounds like GR itself has little to say about the question I am asking and I need to study quantum field theories (!).

In GR, we say that when we drop an apple, it is following a geodesic path. (One way of mathematially describing a geodesic path is a path that maximizes proper time).

So there is no "action at a distance" on the falling apple in GR, there is no "force" pushing it as there is in Newtonian theory. In fact it takes a force to keep the apple from falling - this is provided by the stem of the apple, as long as the stem is intact.

When the stem breaks and the apple falls, there is merely an object that is following a geodesic path.

In flat space-time, geodesics happen to be straight lines. In GR, the concept of "geodesic deviation" encapsulates why objects accelerate relative to each other when each object individually follows a geodesic. For more on the technical side of geodesic deviation and curvature see for instance

http://www.eftaylor.com/pub/chapter2.pdf

especially figure 2, the "ant on an apple" diagram. The ants crawling on the apple all follow straight lines - but they appear to accelerate relative to one another.

Some other shorter sources on the mathematics of curvature and geodesic deviation:

http://en.wikipedia.org/wiki/Geodesic_deviation
http://www.mth.uct.ac.za/omei/gr/chap6/node11.html

 

[jimbobjames]

@masudr: Thanks for that clarification.

@yogi: "Even more puzzling is why two clocks in a uniform acceleration field run at different rates. Where there is a difference in the gravitational force (as is the case near the earth) it would seem there is some physical bases for the phenomena. But where the front clock in a rocket and the rear clock are subjected to identical accelerations, how does the lower clock know it is the lower clock, and vice versa."

Very helpful. But does the clock at the front really tick more quickly than the one at the rear? Or is it that signals from the higher clock reach the lower clock at a higher rate so that it appears to tick more quickly. As observers in a diffferent frame - or mathematicians analyzing the situation - we can readily understand (see) what is going on?

@cesiumfrog: "Not only is there no "action" on the apple"

I know - I was putting out that postulate in order to show that it was not correct and that therefore there needed to be something in contact with the apple.

"the apple does not accelerate towards the ground". Correct - I typed that too quickly.

The question is - do you agree that there is something (field, gravitational wave, gravitons, whatever) in contact with the apple, or do you think there is in fact nothing physical in contact with the apple?

@pervect: Thanks for the links - will check them out.

I know that GR says that the apple follows this path called a geodesic. But just as the ant on the apple is in contact with the apples skin (surface) - and so by simply walking straight it follows what appears to us to be a curved path - so too is the apple in contact with spacetime (the manifold) locally. The nature of that contact is what I am trying to get to.

 

[pervect]

The nature of the contact in essence is that classically, an object has a position given by four coordinates, for example x,y,z, and t.

Note that though this may seem blindingly obvious, it applies only to classical objects (and not, for instance, to quantum objects which have a more complex description in terms of a wavefunction).

For more advanced details, you really need to study manifolds, but one of the most basic properties of manifolds is that they do have "charts" (which may be valid in only a restricted region) which give every point coordinates. In the case of GR, space-time is modeled by a 4-d manifold, so every point has 4 coordinates.

We not only need manifolds - but we need manifolds with a metric. This basically says that we can define the "distance" between objects. In relativity, it turns out that distance and time are not really fundamental, in the sense that they vary with the observer.

What is really fundamental is an entity known as the Lorentz interval. So we really define not the "distance" between points, but the Lorentz interval between them.

There are several ways of describing how objects move through space-time, one of the simplest is the "principle of least action", also known as "the principle of maximal aging".

This is of course a mathematical idea, so if you are looking for gears, or levers, or fluids or something physical that "make" objects move you might be disapointed. But of course one can argue (with a lot of merit) that gears, levers, fluids, etc are examples of physical law, a consequence of reality rather than a description of it. (They can be somewhat useful, but generally such models are usually just rough approximations).

 

[jimbobjames]

OK, here is what I understand so far: If you let go of the apple at a point P, we can find a local (free-fall) reference frame there such that the metric (at that point) is that of special relativity and the first derivative of the metric wrt each of the 4 coordinates is zero. (Einstein would say if you fall then there is no gravity, no force). Local experiments over a very short time in that frame would give the same results as those of SR.

By looking at the coordinate transformation from another frame (say as an observer on the surface of the earth) we arrive at the geodesic equation. The metric connections (Christoffel symbols) in that equation depend only on the metric and its derivatives and when they are non-zero we are dealing with a curved manifold.

In the weak field of the Earth it turns out that only the 0,0 component of the metric differs from that of flat space and GR reduces to Newtonian gravity (due to the curvature of the time part of spacetime).

And so, armed with the equivalence principle and some basic calculus on manifolds, we get a lot of results quickly. The rest of Einsteins field equations has to do with showing how energy and mass effect the metric and its (second) derivatives. So you start with the Stress-Energy tensor, work out the metric and then work out the paths.

Maybe my problem lies in thinking there is something locally curved at the apple (the local spacetime) telling it how to move, when in reality it only appears to be so because I am standing on an accelerated frame? - ie from my perspective as an accelerated observer I see (and calculate) the apple's local spacetime to be curved, whereas it is actually flat. I am getting closer now?

Spacetime at the apple is flat, the curvature arises only from my perspective as an accelerated observer?.

And then the question would reduce to - why am I accelerating toward the apple?!? - OK, because the material in the Earth is being PUSHED by the material below it and so on until we reach the center of the earth. But if the guy on the other side of the world is accelerating towards his apple, then is'nt the world expanding and pushing at our feet?

Do you know of any good animations (maybe in flash or something) of the dynamics of spacetime around a planet? It would be great to see how striaght lines in that curved space send the apple to the ground (or the ground to the apple).

Pervect, is it your belief (understanding) that there is in fact nothing physical in contact with the apple relaying to it the local curvature? Can you think of an experiment to check if there is anything physical in direct contact with it? (I can't, except maybe beaming something of extremely high energy at that space and checking if whatever is there makes itself known to us?).

Cheers.

 

[masudr]

maybe beaming something of extremely high energy at that space and checking if whatever is there makes itself known to us?

That's quite tricky, because in a quantum universe, you start probing the vacuum.

 

[MeJennifer]

Maybe my problem lies in thinking there is something locally curved at the apple (the local spacetime) telling it how to move, when in reality it only appears to be so because I am standing on an accelerated frame? - ie from my perspective as an accelerated observer I see (and calculate) the apple's local spacetime to be curved, whereas it is actually flat. I am getting closer now?

No, both the apple and you are in curved space-time.

 

[cesiumfrog]

@cesiumfrog: "Not only is there no "action" on the apple"

I know - I was putting out that postulate in order to show that it was not correct and that therefore there needed to be something in contact with the apple.

No, I'm saying that your argument is wrong, and therefore there doesn't need to be (and there isn't) anything in contact with the apple.

 

[pervect]

OK, here is what I understand so far: If you let go of the apple at a point P, we can find a local (free-fall) reference frame there such that the metric (at that point) is that of special relativity and the first derivative of the metric wrt each of the 4 coordinates is zero. (Einstein would say if you fall then there is no gravity, no force). Local experiments over a very short time in that frame would give the same results as those of SR.

By looking at the coordinate transformation from another frame (say as an observer on the surface of the earth) we arrive at the geodesic equation. The metric connections (Christoffel symbols) in that equation depend only on the metric and its derivatives and when they are non-zero we are dealing with a curved manifold.

Overall this is a very good summary. Note that for example, an observer in an accelerating rocketship has non-zero Christoffel symbols (which can be derived from the metric and its first derivative) but a zero Riemann tensor (which can be derived from the metric and its first and second derivatives). For some purposes it is convenient to define a manifold as curved only when the Riemann is non-zero, and flat when the Riemann is zero. With such a definition, an accelerated observer in flat space-time can say that he has non-zero Christoffel symbols and a "flat space-time" (i.e. zero Riemann tensor). So by this defintion, non-zero Christoffel symbols don't imply non-flatness.

In other words, switching to an accelerated frame makes the Christoffel symbols non-zero and the associated frame non-inertial, but by some common defintions of "curvature" (i.e. the nonzero Riemann tensor definition of curvature) it may not mean that space-time becomes curved.

In some other cases, though, writers do talk about the manifold associated with an accelerating observer as if it were curved even though the Riemann is zero. For instance, many textbooks say that gravitational redshift is evidence of "curvature". But this means that the authors of such textbooks must be adopting the "nonzero Christoffel" definition of curvature, because an accelerating observer has a zero Riemann.

In the weak field of the Earth it turns out that only the 0,0 component of the metric differs from that of flat space and GR reduces to Newtonian gravity (due to the curvature of the time part of spacetime).

Yes.

And so, armed with the equivalence principle and some basic calculus on manifolds, we get a lot of results quickly. The rest of Einsteins field equations has to do with showing how energy and mass effect the metric and its (second) derivatives. So you start with the Stress-Energy tensor, work out the metric and then work out the paths.

Maybe my problem lies in thinking there is something locally curved at the apple (the local spacetime) telling it how to move, when in reality it only appears to be so because I am standing on an accelerated frame? - ie from my perspective as an accelerated observer I see (and calculate) the apple's local spacetime to be curved, whereas it is actually flat. I am getting closer now?

Spacetime at the apple is flat, the curvature arises only from my perspective as an accelerated observer?.

There is much confusion in the literature, especially the popularized literature, about what the term "curvature" really means as I remarked earlier. The technical point is that an observer following a geodesic in a Minkowski space-time has locally zero Christoffel symbols, while an accelerating observer in the identical Minkowski space-time has non-zero Christoffel symbols. If one uses the Riemann definition of curvature, however, one would describe the Minkowski space-time as being flat, independent of the observer, so that the space-time is flat both for the accelerating and non-accelerating observer.

Using the non-zero Christoffel symbol definition of curvature would mean that whether or not space-time was curved depended on the observer. This is a rather confusing usage, but some textbooks do it anyway, especially when they talk about the equivalence principle.

Mostly, what I can say is "be aware of the issue".

Pervect, is it your belief (understanding) that there is in fact nothing physical in contact with the apple relaying to it the local curvature? Can you think of an experiment to check if there is anything physical in direct contact with it? (I can't, except maybe beaming something of extremely high energy at that space and checking if whatever is there makes itself known to us?).

Cheers.

With my GR hat on, I would say that there is only space-time. When you start to get into quantum mechanics, though, "empty" space starts to become full of virtual particles, so with a quantum hat on one would probably not take the same perspective about the "emptiness" of the vacuum.

It is also a mystery why cosmological observations suggest that space-time may have a small "cosmological constant" which could be interpreted as "not being empty".

One can separate this issue out by calling the cosmological constant "dark energy". So empty space may possibly still contain this "dark energy". However, the origins of this "dark energy" aren't really well understood, and we have only some cosmological arguments at this point to support its existence. (Though most people do seem to be convinced that the cosmology here is good.)

 

[MeJennifer]

Good summary on curvature Pervect!

I wish everybody would stop calling space-time curved in cases where the Riemann tensor is zero, it gives great confusion and a lot of wasted time in explaining.

Many mistake, or even attempt to equate, the properties of hyperbolic space for curvature.

 

[yogi]

@masudr: Thanks for that clarification.

@yogi: "Even more puzzling is why two clocks in a uniform acceleration field run at different rates. Where there is a difference in the gravitational force (as is the case near the earth) it would seem there is some physical bases for the phenomena. But where the front clock in a rocket and the rear clock are subjected to identical accelerations, how does the lower clock know it is the lower clock, and vice versa."

Very helpful. But does the clock at the front really tick more quickly than the one at the rear? Or is it that signals from the higher clock reach the lower clock at a higher rate so that it appears to tick more quickly. As observers in a diffferent frame - or mathematicians analyzing the situation - we can readily understand (see) what is going on?

[/I].

Y
That is always a bothersome question - from the standpoint of GPS and Hafele, the effect is real - the clock at the lower gravitational potential actually accumulates less time when compared to the upper clock - but this doesn't comport with the way the effect is derived - for example the thought experiment of sending signals that are spread further apart because the spacing of the received pulses is altered by the additional distance that each pulse must travel during the acceleration - this clearly suggests an apparent (observed) difference between two clocks that are both running at the same rate ...an effect that should disappear with no residual measureable difference between the two clocks after the acceleration ends and the clocks are brought together and compared - so it seems that the thought experiment gives the right answer for the wrong reason .. if the rate of the lower clock is actually altered.

 

[jimbobjames]

Thanks a lot pervect for pointing out that distinction between the non-zero Christoffel symbol versus the non-zero Riemann definitions of curvature. I was'nt aware of that.

Adopting the Riemann definition, the spacetime at the apple then is also curved (thanks MyJennifer). Can we speak of an absolute curvature at that point then? ie although the components of the Riemann may change depending on your choice of frame, there is some invariant scalar which describes the curvature at that point - let me guess, the Ricci?

I wonder if we spent most of our time in a free-fall frame would this all be more intuitive - that maybe the difficulty I am having arises by virtue of having evolved (and grown up) on an accelerated frame.

Anyway, I am really fascinated by the apple falling to the ground (or vice versa, cesiumfrog) when there appears to be nothing between the two. I used to ask myself - how does the apple know that the Earth is where it is? The answer I am getting is that it does'nt, that it only knows ("sees", "feels") its local spacetime and happily follows a straight line along it (its geodesic).

So then I wanted to understand more about that contact between the apple and the local spacetime and have come to a bit of a dead end on that one. It seems the best I can get from GR is that there are mathematical objects there (the Riemann, the Christoffel symbols, the metric) which are enough to calculate which way the apple will move. (And because I cannot devise an experiment to probe that local spacetime, the question is relegated to the philosophical. Unless maybe I put on a quantum hat )

 

[pervect]

T
Adopting the Riemann definition, the spacetime at the apple then is also curved (thanks MyJennifer). Can we speak of an absolute curvature at that point then? ie although the components of the Riemann may change depending on your choice of frame, there is some invariant scalar which describes the curvature at that point - let me guess, the Ricci?

The Riemann can be regarded as a geometric object which is independent of coordinates (as can any tensor).

I believe that there are a number of invariant scalars that one can compute from the Riemann, such as the Ricci scalar, the Kretschmann scalar, and others, too.

see for example
http://en.wikipedia.org/wiki/Kretschmann_invariant

 

[jimbobjames]

"The Riemann can be regarded as a geometric object which is independent of coordinates (as can any tensor)."

Yes, but if you choose to determine the components related to a particular coordinate system, those components will vary depending upon your choice of coordinate system. The geometric object is independent, yes, but not the components which depend on the choice of coordinates.

That's what I was referring to when I wrote "the components of the Riemann may change depending on your choice of frame".

Had'nt heard of the Kretschman - thanks again.

 

[MeJennifer]

So then I wanted to understand more about that contact between the apple and the local spacetime and have come to a bit of a dead end on that one.

But why do you think there ought to to be some kind of contact?

Think about it purely relational: have a system with n wordlines that have mass and energy, observe how their metric distances vary over coordinate time.
Now when you try to map all those distances on a flat manifold you can find out that that is impossible, instead you need a curved manifold.
But is the curved manifold the cause or the result?

Furthermore, could we (in principle and classically) prove it is a manifold?
All we can measure is a topological mesh that does not fit in a flat 4-dimensional grid. The interpolation of what is "in between" has no physical significance, since there are no particles there, and interestingly, if we place a particle in-between, the prior interpreted value must change since all particles either have mass or energy.

 

[pervect]

"The Riemann can be regarded as a geometric object which is independent of coordinates (as can any tensor)."

Yes, but if you choose to determine the components related to a particular coordinate system, those components will vary depending upon your choice of coordinate system. The geometric object is independent, yes, but not the components which depend on the choice of coordinates.

No argument here, that's totally correct.

 

[jimbobjames]

@pervect and others: I'm really glad I stumbled upon this forum. I've been bothered by these questions for years and you're helping me get to the answers.

General Relativity is a classical field theory, right?

The question I was asking is:

"What is the nature of the physical entity which is in direct contact with that particle"

Is'nt the answer then simply - the real physical entity in contact with the particle is the field (described/modeled by the metric, and it's derivatives)?

In terms of an experiment to determine its existence (at least indirectly): You could determine its real existence as well as its characteristics by placing other test particles - assume negligable mass - in the vicinity of the original particle and observing the effect of the field on those test particles - (I know this may seem blindingly obvious but I am trying to address pervect's request to think of an experiment to show its existence).

And further (do correct me if I'm wrong here): That field has an absolute character, independent of your choice of coordinate system - and it is against this field (background) that we can measure / determine if something is really (absolutely) accelerating or not?

@MyJennifer - "Now when you try to map all those distances on a flat manifold you can find out that that is impossible, instead you need a curved manifold." That helps a lot - Thanks!

 

[pmb_phy]

@Pete: Thanks. I didnt mention this, but I have studied GR in my spare time and would say I have a fairly good grasp of the mathematics. Perhaps my favorite introduction to GR is that by Dirac who gets straight to the point - I think I was pleasantly surprised that I could follow his derivations.

Since you understand the math then you understand what a manifold is. A manifold is not a physical entity but a mathmatical construct used to describe nature, in this case it describes tidal forces. It is the manifold itself that is curved. In relativity the manifold is the spacetime manifold. Do you disagree with this?

Best wishes

Pete

 

[jimbobjames]

"Do you disagree with this?"

No, not at all. Thanks.

 

[pervect]

@pervect and other: I'm really glad I stumbled upon this forum. I've been bothered by these questions for years and you're helping me get to the answers.

General Relativity is a classical field theory, right?

The question I was asking is:

"What is the nature of the physical entity which is in direct contact with that particle"

Is'nt the answer then simply - the real physical entity in contact with the particle is the field (described/modeled by the metric, and it's derivatives)?

"Real" and "physical" are a bit hard to define, but yes, considered as a field theory, the Einstein-Hilbert action is just the Ricci scalar, which can be derived from the metric and it's second derivatives.

http://en.wikipedia.org/wiki/Einstein-Hilbert_action

Extremizing this action basically gives you the field equations of GR in the normal way.

In terms of an experiment to determine its existence (at least indirectly): You could determine its real existence as well as its characteristics by placing other test particles - assume negligable mass - in the vicinity of the original particle and observing the effect of the field on those test particles - (I know this may seem blindingly obvious but I am trying to address pervect's request to think of an experiment to show its existence).

You can measure the Riemann tensor directly by the deflection of test particles. The various components of the Riemann will appear to be tidal forces (though I'm of the opinion that one should add for safety's sake that the tidal forces should be measured in a non-accelerating, non-rotating frame).

There is even a breakdown of the Riemann into velocity independent parts (sometimes called "electric", though it has nothing directly to do with electromagnetism), and velocity dependent parts (sometimes called magnetic).

And further (do correct me if I'm wrong here): That field has an absolute character, independent of your choice of coordinate system - and it is against this field (background) that we can measure / determine if something is really (absolutely) accelerating or not?

The metric has an absolute character, and so does the Riemann, in the sense that they arae both geometric entities.

As we discussed earlier, the Christoffel symbols can be derived from the metric, and they tell you whether or not you are accelerating. (The Christoffel symbols aren't a tensor, though.)
[edit - delete]

So the short version is that the "field" of GR considered as a field theory is just the geometry of space-time, as given by the metric.

 

[jimbobjames]

Thanks a lot pervect.

It must have been a beautiful thing to arrive at the field equations separately and independently by extremizing an action.

It's somewhat ironic that some of the key objects of study in General Relativity - like the metric and the Riemann - are in fact absolute in character.

When I look at what otherwise appears to be empty space, I am more confident now that the space is not empty but alive with the metric tensor field of GR. I can test that it exists by placing test particles there and seeing the effect of the field on the particles.

Now a charged particle placed in the manifold is affected also by the em field (and we can use the em field tensor to do the calculations). Where do the similarities between the em field and the field of GR stop (apart from the fact that there is only positive mass making gravitational effects cumulative, and that electromagnetism has been successfully quantized (and indeed unified with the other forces)? Are there other fundamental differences? Perhaps that gravity lends itself to a complete geometrical interpretation unlike electromagnetism?

I mean my question about the fundamental reality ("physical nature") of the metric field could have equally been applied to other fields where we can see the effects of the field (say on iron filings around a magnet) and where we have an excellent mathematical model which predicts the motions of test (charged) particles in the field, but as pervect says, there are no gears and cogs there.

It's almost as if all of spacetime - even the apparently empty subspace - is completely filled with mathematical objects (we call fields) which, through physical laws, interact with ponderable matter.

(Apologies for the philosophical tone, I just can't think of any other way to express it).

Two related questions:

1) Do we know the overall shape/topology of our spacetime? How accurate is the analogy with the 2-d surface of an expanding baloon (to represent a 3-d spatial slice through time) - with some bumps and valleys near dense objects? (Or is this a massive over-simplification?).

2) Is it possible that the total energy in the manifold is zero? I had read some time back that in Newtonian Gravity there could be a zero sum universe given that the energy in the field is negative, and could exactly balance the positive energy of the masses - I was wondering if that still holds in GR and if the idea of a zero sum energy universe is still given any attention?

Thanks.

 

[MeJennifer]

It's somewhat ironic that some of the key objects of study in General Relativity - like the metric and the Riemann - are in fact absolute in character.

I beg to differ.

The metric of space-time completely depends on the influence that each particle has on space-time in a non linear way.

When I look at what otherwise appears to be empty space, I am more confident now that the space is not empty but alive with the metric tensor field of GR. I can test that it exists by placing test particles there and seeing the effect of the field on the particles.

But in reality there are no test particles and if you interject a particle into the "soup", the metric will change!

This element to me is the essense of the theory of relativity.

For instance take the Schwarzschild solution. Is it an exact solution? Yes, if there are no other particles!
As soon as we have one other particle the solution is not longer exact but at best an approximation.

 

[coalquay404]

I wish everybody would stop calling space-time curved in cases where the Riemann tensor is zero, it gives great confusion and a lot of wasted time in explaining.

Simply because a physical observable has value zero does not mean that it's not an observable.

 

[jimbobjames]

@MeJennifer: "The metric of space-time completely depends on the influence that each particle has on space-time in a non linear way."

I was referring to absolute in the sense that the metric and Riemann are geometric objects which are independent of any choice of reference frame.

And yes, you are right - even test particles with negligable mass will still have an effect on the field, however vanishingly small that effect may be.

 

[masudr]

But in reality there are no test particles and if you interject a particle into the "soup", the metric will change!

This element to me is the essense of the theory of relativity.

We use test particles in electrodynamics too! Even in non-relativistic field theories. We know that there is no such thing: test particles are defined by a limiting process in order to help understand nature better. No one is claiming a test particle will not affect the field in question (whether it be the gravitational, Newton gravitational, electromagnetic etc.)

 

[MeJennifer]

No one is claiming a test particle will not affect the field in question (whether it be the gravitational, Newton gravitational, electromagnetic etc.)

And I was not implying that.

 

[MeJennifer]

I was referring to absolute in the sense that the metric and Riemann are geometric objects which are independent of any choice of reference frame.

Of course they are, they are indeed mathematical objects, e.g. abstractions.

General relativity is a relational theory, the manifold has no independent existence, it simply represents the relationships of all space-time events. Or as Rovelli wrote: "We can say that GR is the discovery that there is no spacetime at all".

Note that it is impossible to separate any kind of background on which events happen in general relativity!

With regards to the manifold, see for instance what Rovelli writes in Quantum Gravity:

" In the mathematical formalism of GR we utilize the "spacetime" manifold M, coordinated by x. However, a state of the universe does not correspond to a configuration of fields on M. It corresponds to an equivalence class of field configurations under active diffeomorphisms. An active diffeomorphism changes the localization of the field on M by dragging it around. Therefore localization on M is just gauge: it is physically irrelevant.
In fact, M itself has no physical interpretation, it is just a mathematical device, a gauge artifact. Pre-general-relativistic coordinates x^u design points of the physical spacetime manifold "where" things happen...; in GR there is nothing of the sort. M cannot be interpreted as the "set" of physical events, or physical spacetime points "where" the fields take value. It is meaningless to ask whether or not the gravitational field is flat around the point A of M, because there is no physical entity "spacetime point A". Contrary to Newton and Minkowski, there are no spacetime points at all. The Newtonian notions of space and time have disappeared. "

Pretty profound right?

 

[jimbobjames]

@MeJennifer: Thanks a lot for pointing me to these ideas from Rovelli.

From the piece you quoted however, I see no contradiction with what I was saying above.

"localization on M is just gauge: it is physically irrelevant"

and

"M itself has no physical interpretation, it is just a mathematical device"

Above I was coming to the conclusion that the physically significant (ie very real) object was the field (modelled mathematically by the metric and its derivatives) and not necessarily the manifold.

The very real existence of that field could be demonstrated by the instantaneous effect it would have on test particles, which you might introduce into the field, even at large spatial separations from the source of the field.

Were there nothing real (physical) at the location of the experiment - which would otherwise appear to be "empty" space - then there would be no explanation for the relative acceleration of the test particle along a line directly to the centre of mass of the source (and not in any other arbitrary direction).

(I admit that I have only carried out the experiment myself at much shorter distances however )